Scaling limits of the Chern-Simons-Higgs energy

We continue our study in [16] of the Gamma limit of the Abelian ChernSimonsHiggs energy G_csh := 1/2 ∫_U |∇_Aε u_ε|² + μ_ε²/4 |curl A_ε - h _ex|²/|u_ε|² + 1/ε² |u_ε|² ( 1 - |u _ε|²)² dx on a bounded, simply connected, two-dimensional domain where ε → 0 and μ(ε) → μ ∈ [0, +∞]. Under the critical scaling, G(csh) ≈ |log ε|² , we establish the Gamma limit when μ ∈ (0,+∞], and as a consequence, we are able to compute the first critical field H"1 = H"1(U,μ) for the nucleation of a vortex. Finally, we show failure of Gamma convergence when μ(μ) → 0 (this includes the self-dual case). The method entails estimating in certain weak topologies the Jacobian J(u(ε)) = det(∇ u(ε)) in terms of the ChernSimonsHiggs energy E(csh).